# Tomoyuki ICHIBA

### Associate Professor, Department of Statistics & Applied Probability and Center for Financial Mathematics and Actuarial Research, and Mathematics in College of Creative Studies

### at University of California Santa Barbara

## Current Research

**Financial Markets with Discontinuities**(NSF DMS-1313373)

This project studies some topics in the system of Stochastic Differential Equations with possibly discontinuous and degenerate diffusion coefficients with applications to stochastic portfolio management and optimization problems. Here the coefficients of the equations are functions of the states and determine the stochastic behavior of the system. The coefficients, the initial states and the stochastic noise of the system are main inputs of the model. The output of the system is the stochastic process which can be construed as the dynamics of the financial system. The long-term goals of the proposed activity are two-folds: (i) to defend against financial crises and (ii) to provide efficient investment strategies in the financial markets.

**Collisions of Brownian particles**: Let us consider n tiny Brownian particles which move in the real line and whose dynamics are described by Stochastic Differential Equations with local drifts and local volatility (variance/covariance) coefficients. On the probability space induced by the stochastic differential equations these Brownian particles may collide in the real line. How do these coefficients determine the colliding behaviors of the particles?- When do three or more particles collide?
- What kind of conditions on the coefficients imply colliding or non-colliding behaviors?
- If they collide, how do they move right after the collision?

**Stochastic Portfolio Theory**: The financial equity market has many aspects in its dynamics and stochastic properties. What kind of mathematics can one use in order to understand the market structure? How can our understandings be applied to the better portfolio management? ``Stochastic Portfolio Theory is a novel mathematical framework for analyzing portfolio behavior and equity market structure." [In Preface of Applications of Mathematics Stochastic Modeling and Applied Probability series 48 ``Stochastic Portfolio Theory" (E. Robert Fernholz (2002))]**Strong/Weak Solutions of Rank-based Continuous Markov Processes**: Let us consider n-dimensional stochastic differential equations where the coefficient functions depend on the rankings of the coordinate process corresponding to the equations. Is the system of the stochastic differential equations solvable? If so, in what sense are they solvable, weakly or strongly? This class of diffusions has some interesting behaviors and some applications to Stochastic Portfolio Theory.**Semimartingale Local Times**: The local time process accumulated at a site by a semimartingale measures the intensity of occupation of the semimartingale in the neighborhood of the site. It appears in the above the problems of colliding Brownian motions and the rank-based continuous Markov processes. The main issues are characterization of the local times with various applications.**Stability and Instability of Interbank-lending**: We suffer from the major financial crisis in 2007. In the financial crises flows of money becomes quite important. The flows can be seen as the consequence of various financial activities. What kind of activities can cause the crises? How can one analyze the flows mathematically? Viewing the whole bank activities in an interacting particle system, we analyze the stability and instability of the system.**Time Reversal**: For some class of diffusions it is known that there are some relationship between the forward process and backward process in time. Under some conditions the forward and backward processes match in distribution. We study some applications of time reversal.

## Preprints

(1) ``Large deviations for interacting Bessel-like processes and applications to systemic risk" (with M. Shkolnikov) arXiv1303.3061

(2) ``Stochastic integral equations for Walsh semimartingales" (with Ioannis Karatzas, Vilmos Prokaj and Minghan Yan) arXiv: 1505.02504

(3) ``Yet another condition for absence of collisions for competing Brownian particles" (with Andrey Sarantsev) arXiv: 1608.07220

## Recently Published Papers

[11] ``Skew-unfolding the Skorokhod reflection of a continuous semimartingale" *Stochastic Analysis and Applications 2014 * In Honor of Terry Lyons. Springer Proceedings in Mathematics & Statistics (2014) 349-376 (with Ioannis Karatzas) arXiv: 1404.4662.

[10] ``Planar diffusions with rank-based characteristics and perturbed Tanaka equations" *Probability Theory and Related Fields* **156** (2013) 343-374; a full version with additional topics of Transition probabilities, Time reversibility and Maximality is also available in
(with E. R. Fernholz, V. Prokaj and I. Karatzas) arXiv:1108.3992 .

[9] ``Strong solutions of stochastic equations with rank-based coefficients" * Probability Theory and Related Fields* ** 156 ** (2013) 229-248 (with I. Karatzas and M. Shkolnikov) arXiv1109.3823 .

[8] ``Diffusions with rank-based characteristics and values in the nonnegative quadrant" *Bernoulli* **19** (2013) 2455-2493 (with Ioannis Karatzas and Vilmos Prokaj) arXiv:1202.0036.

[7] ``Convergence rates for rank-based models with applications to portfolio theory"
*Probability Theory and Related Fields* ** 156** (2013) 415-448 (with S. Pal and M. Shkolnikov) arXiv:1108.0384.

[6] ``A second-order stock market model" *Annals of Finance* **9** (2013), 439-454 (with R. Fernholz and I. Karatzas) arXiv:1302.3870.

[5] ``Two Brownian particles with rank-based characteristics and skew-elastic collisions'' *Stochastic Processes and their Applications*** 123 ** (2013) 2999-3026 (with E. Robert Fernholz and Ioannis Karatzas) arXiv:1206.4350 .

[4] ``Stability in a model of inter-bank lending" *SIAM Journal of Financial Mathematics* ** 4** (2013) 784-803 (with J-P. Fouque).

[3] ``Hybrid Atlas models"
*Annals of Applied Probability* ** 21** no 2. (2011), 609-644.
(with R. Fernholz, I. Karatzas, A. Banner and V. Papathanakos) arXiv:0909.0065.

[2] ``Collisions of Brownian particles"
*Annals of Applied Probability* ** 20** no 3. (2010), 951-977 (with I. Karatzas) arXiv:0810.2149.

[1] ``Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation"
* Journal of Applied Probability * ** 48 ** no 3. (2011) 699-712 (with Constantinos Kardaraz) arXiv:1008.1326.

## Dissertation

``Topics in multi-dimensional diffusions: attainability, reflection, ergodicity and rankings'' Dissertation Columbia University. 2009. [924kb].

## Conference Proceedings

[C2] `` Folding and Skew-Unfolding of One-dimensional Continuous Semimartingales" *RIMS Kokyuroku No 1952: Symposium on Probability Theory, Kyoto 2014 *

[C1] ``On planar rank-based diffusions with skew-elastic collisions" *RIMS Kokyuroku No 1903: Symposium on Probability Theory, Kyoto 2013 *

## Posters

``Stochastic Integral Equations for Walsh Semimartingales" IMA Special Workshop: Reflected Brownian motions, Stochastic Networks and their applications at University of Minnesota (June 2015)

``Strong/Weak Solutions of 2D Diffusions with Rank-based Characterisits" Probabiliy, Control and Finance at Columbia University (May 2012)

### Miscellaneous Papers of Statistics in Sports, Biology and Actuarial Science

(S4) ``Estimating the effect of the red card in soccer: when to commit an offense in exchange for preventing a goal opportunity''
* Journal of Quantitative Analysis in Sports* ** 5**, no. 1 / 2009. (with Jan Vecer and Frantisek Kopriva).

(S3)``On probabilistic excitement of sports games.'' * Journal of Quantitative Analysis in Sports*
** 3**, no. 6 / 2007. (with Jan Vecer and Mladen Laudanovic).

(S2) ``Assessing substitution variation across sites in grass chloroplast DNA.''
* Journal of Molecular Evolution* ** 64**, no. 6 pp. 605-613 / June, 2007.
(with Tian Zheng and Brian Morton) .

(S1) ``On multi-period statistical risk management methods and equity-linked life insurance.''
* Journal of the Japan Statistical Society*. Japanese issue ** 35**, no. 2
pp. 103-123. / 2006. abstract.
(with Naoto Kunitomo).